#34205391

order	20224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₇₉ ⋊ #13021

Subgroups

	subgroup	G/H
center	C₄	#19098695
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting, prime radical	C₇₉×#3414	C₂
socle	C₂×C₇₉	#1879
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#3414	D₇₉
hypocenter	C₇₉	#13021
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #3414

462 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 143
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 3 generators on 143 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(80,83)(81,90)(82,97)(84,105)(85,103)(86,112)(87,114)(88,107)(89,115)(91,120)(92,119)(93,127)(94,129)(95,122)(96,130)(98,136)(99,132)(100,137)(101,135)(102,125)(104,110)(106,109)(108,139)(111,133)(113,134)(116,138)(117,131)(118,142)(121,124)(123,143)(126,140)(128,141);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,82,87,113,88,101,109,100)(81,89,94,128,95,118,124,117)(83,97,114,134,107,135,106,137)(84,108,86,98,110,99,85,111)(90,115,129,141,122,142,121,131)(91,123,93,96,125,116,92,126)(102,138,119,140,120,143,127,130)(103,133,105,139,112,136,104,132);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81,86,93,87,94,110,125,88,95,85,92,109,124,84,91)(82,96,99,118,100,123,98,128,101,126,108,89,113,116,111,117)(83,102,105,129,106,127,103,90,107,120,104,121,114,119,112,122)(97,131,133,138,134,115,139,140,135,141,136,143,137,142,132,130)

Polycyclic presentation:

2^8,79|a^2=f,b^2=dfg,d^2=gh,e^2=g,f^2=g,g^2=h,b^a=bd,c^a=ce,i^a=i^78,d^b=dg,e^b=egh,f^b=fg,g^b=gh,i^b=i^78,d^c=dgh,e^c=egh,f^c=fgh,g^c=gh

Conjugacy classes & character table

1294 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 86690);